Erdős-Ko-Rado theorem for irreducible imprimitive reflection groups

Abstract

Let Ω be a finite set, and let G be a permutation group on Ω. A subset H of G is called intersecting if for any σ, π ∈ H, they agree on at least one point. We show that a maximal intersecting subset of an irreducible imprimitive reflection group G(m, p, n) is a coset of the stabilizer of a point in {1, . . . , n} provided n is sufficiently large.

Key words： Erdős-Ko-Rado theorem; representation theory; imprimitive reflection groups

References

1 Ariki S. Representation theory of a Hecke algebra of G(r, p, n). J Algebra, 1995, 177: 164-185

2 Ariki S, Koike K. A Hecke algebra of (ℤ/rℤ)Sn and construction of its irreducible representations. Adv Math, 1994, 106: 216-243

3 Birkhoff G. Three observations on linear algebra. Univ Nac Tucumisán Revista A, 1946, 5: 147-151

4 Cameron P J, Ku C Y. Intersecting families of permutations. European J Combin, 2003, 24(7): 881-890

5 Deza M, Frankl P. On the maximum number of permutations with given maximal or minimal distance. J Combin Theory Ser A, 1977, 22: 352-360

6 Diaconis P, Shahshahani M. Generating a random permutation with random transpositions. Zeit Für Wahrscheinlichkeitstheorie, 1981, 57: 159-179

7 Ellis D. A proof of the Deza-Frankl conjecture. arXiv: 0807.3118, 2008

8 Ellis D, Friedgut E, Pilpel H. Intersecting families of permutations. J Amer Math Soc, 2011, 24(3): 649-682

9 Erdős P, Ko C, Rado R. Intersection theorems for systems of finite sets. Quart J Math Oxford Ser, 1961, 12(2): 313-320

10 Godsil C, Meagher K. A new proof of the Erdős-Ko-Rado theorem for intersecting families of permutations. European J Combin, 2009, 29: 404-414

11 Halverson T, Ram A. Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection group G(r, p, n). Can J Math, 1998, 50(1): 167-192

12 James G, Kerber A. The representation theory of the symmetric group. Encyclopedia of Mathematics and its Applications, 1981, 16

13 Larose B, Malvenuto C. Stable sets of maximal size in Kneser-type graphs. European J Combin, 2004, 25(5): 657-673

14 Li Y S, Wang J. Erdős-Ko-Rado-type theorems for colored sets. Electron J Combin, 2007, 14(1)

15 Read E W. On the finite imprimitive unitary reflection groups. J Algebra, 1977, 45(2): 439-452

16 Serre J-P. Linear Representations of Finite Groups. Berlin: Springer-Verlag, 1977

17 Shephard G C, Todd J A. Finite unitary reflection groups. Can J Math, 1954, 6: 274-304

18 Wang J, Zhang S J. An Erdős-Ko-Rado-type theorem in Coxeter groups. European J Combin, 2008, 29: 1112-1115